If this is your question, this answer
is wrong.
The question you answered was "what is the probability that a given person is within a given subnet - if completely random?"
To answer your original question, you need to know how many people are on the "network". Assuming a 16-bit network address, 8-bit subnet, and 8-bit host address:
If there are more than 256 people on the same network, which is clearly the case on a University for example, then some of them "will" have to share the same subnet. Easy to see right?..
Of course you can do a involved calculation if needed be. But why bother.. Most networks have over 256 users in them. Then it is guaranteed (probability 1) than some of them will share the same subnet.